I agree about the difficulty of trying to suggest presents, where you do not want to look greedy, and sometimes you know what they would like or you would like to buy them but feel it looks too much to suggest.
So a couple of ideas that might be worth considering. I used to teach in a montessori and we taught them how to pick up bricks etc that treated things gently but also was strengthening for their fingers that would be used for writing later on. So we had lovely wooden bricks etc. You might look at that sort of thing wooden bricks or train or whatever. They are usually more expensive than the plastic ones but do last for a very long time, are very pleasant to touch and use and can be used with other things later on as they grow. Another possibility is to begin a lego collection. In the same way these are good in themselves , but also as you get more of them they can be used for buildings on a train set or to make a whole lego board in years to come. Things like that last for a long time and are still in use long after other things have disappeared.
Then I am a bookworm and have loved reading all my life and encourage all ages to enjoy reading. At this age I suggest a lovely book is Each Peach Pear Plum by Allan Ahlberg. I tend to buy the paperbacks which are cheaper but you could maybe buy the hardback if you wished. It is a lovely book and uses all the nursery rhyme characters in the story. This book I have used with ages up to 5 and 6 where they like to read it to younger children and then there are also classics like The very hungry caterpillar, Meg and Mog, and the Tiger who came to Tea.
As children get older I would give book tokens and a small toy. There is great pleasure in being able to go into a book shop and choose for yourself. As children rarely have money at an early age, you find the adults are doing the choosing and I treasured the lovely feeling of being able to pick what I wanted for myself. Going back to the wooden idea, there is a place near here where they make wooden toys and one of the things they do is make a wooden engine with wooden carriages and each one has a letter on. So you could pick out the letters of their name. Whilst they are little it can be a decoration in their room or on a window ledge and as they are able to play with it it is then their special toy that will be kept . I have done this for several children over the years and also used that for a christening present and have even seen one friends child grow up and he has treasured his train and now it is next to his own sons one on the window ledge ready for using in the future.
Whilst these are all ideas for now, if you have funds available, perhaps you could put an amount each year into a savings account but keep it in your name not theirs. You will know it is for them. Then as the money mounts up I would look towards buying a life membership of the national trust or the RHS in my case as we are all gardeners. It may not be greeted as the greatest gift at the time but my goodness I have met many people over the years who have blessed the gift which lets them visit where they want and as the price goes up it becomes one of the biggest bargains they could have - especially when you see the NT car parks you can use for free!
My granny used to give us christmas gifts but always two childrens classic books as well and when the toys or presents are long forgotten the books she introduced me to and that I loved are always remembered , with some of the books still here and the memory of our peacefully sitting reading side by side is a wonderful thought of her and her great ideas.
No doubt you will have other ideas that will come to you , but rather than seeing it as some sort of competition with the other family members you become the thoughtful granny who gives unusual gifts. Later you can again look forward to giving experiences as birthday gifts so all sorts, a trip to a fair, or a ride on a train to the coast or on a steam train if they enjoy Thomas the tank engine. A little later on being taken out to tea on their own where they can choose the menu, even if it is icecream followed by cake followed by lemonade. Not to be recommended in general but as a one off. Can they visit you and have a lovely morning making pancakes and having a go at tossing one, buy some little suchards eggs and hide them round the garden for an easter egg hunt.
So all this is looking to the pleasure of the future of meeting this new little person on their own merits and enjoying their company. For the moment try to see the kind intention of the mother and perhaps send her a token for a toyshop or write out your own "voucher" offering a trip to somewhere or something when you next get together. As a one year old the little one wont notice one present less on the day and to have an unexpected gift later will be enjoyed and you can have a happy time choosing together. So again in the mother chooses a modest lego set or whatever you could then ask her if she would like the bigger set or something else as well.
Hope I havent gone on too long but that some of these ideas might be of use. You might also have a little notebook and on birthdays and other special days write in it about the gift you gave and perhaps why or what you were thinking of and in years to come they may enjoy reading these notes and understanding the love and care you put into these things.