A three-step tutorial for adding a colourful grain effect to images using Photoshop’s Noise filter
With your image open, add a new layer by pressing Cmd/Ctrl+Shift+N. Go to Edit>Fill and set Contents: Use to Color. Choose a colour from the colour menu that pops up and hit OK. Exit the Fill menu.
Go into the Filter menu and down to Noise>Add Noise. Tick the Monochromatic option in the filter and set this to Gaussian with the Amount set to around 18-20%.
To merge this layer with the image underneath, set its blend mode in the Layers palette to Overlay. The image will have a slight tint, and the grain should show through as well. If the colour is too overpowering, the opacity of the layer can be reduced until the colour is less prominent.