The word ‘speed’ is used here in an archaic sense, but one that you should be easily able to find in a dictionary, for example:
speed, v. 6.a. To further or assist (a person); to cause to succeed or prosper
Oxford English Dictionary
Today this sense mostly survives in stock phrases like ‘God speed’, meaning ‘(may) God assist (you)’.
When Bombadil suggests that ‘the morning will speed you’ what he means is that the hobbits will find it easier to continue on their journey in the morning, when it is light. The attribution of this assistance to the ‘morning’ is a kind of (mild) personification. This is in character for Bombadil, who is prone to speaking of natural objects as if persons, for example,
“Sun won’t show her face much today, I’m thinking.”
J. R. R. Tolkien (1954). The Fellowship of the Ring, book I, chapter 7. London: George Allen & Unwin.