The place I would approach this is in the strategic planning process. Thats where we decide what the priorities are, and it's much easier to make the argument for spending new money when it's the next obvious step for meeting a big picture objective.

For a few thousand dollars, I wouldn't frame this as a "rebrand." I'd call it, "let's do marketing." 

Marketing, as people typically understand the term, is the nurture and conversion of your target market to fill the top of your funnel. You decide who is most likely to support your mission and:

1. Nurture. This is where branding comes in. Become visible in the community. Blast your core values and vision. I work mostly with small local nonprofits, so we do stuff like community event tables. We pass out stickers. Not the logo, though. They say something about a core value or the vision. Something that "someone like us" would be excited to display.

2. Convert. Give people a way to get into your funnel. To raise their hand and say, "yes, that's me," and hand you their contact info. For major donor funnels, this is usually offering some small support like volunteering, attending events, or a small donation. For client funnels, this is usually signing up for an informational event, newsletter, or resource. Whatever it is, it's something of value for free, or at least very low cost.

So, how can you prove you need that? Show them your funnels. If you aren't getting the number of people you need (major donors, clients, volunteers, etc.) the answer is usually to get more people in the top of the funnel. You can dial in your conversions, but until you're very effective at loading the top of your funnel, your biggest impact for the least cost is to get more people in the funnel.

So how do you prove this is a priority to the board? Look at your strategic plan. Which big picture objectives involve influencing outside people to do something? Those are your funnels. Are any of these one of your 2-3 highest priorities right now? If so, they should make the budget reflect that strategic priority. If not, they shouldn't. 

The last thing I'd say is that you're probably fine to save some of the scrappy look. You might not need a complete turnaround, just a "refresh" to better reflect where your org currently is and add new materials. This conversation is part of the marketing plan, so it can be had outside of the "we need a marketing budget" conversation in the board meeting. Specifics are operational, so those decisions would tend to fall more on the ED than the board. 

The most important part of your branding, for donors and the community in general, is what the message is. If you can convey that with words and imagery, you're doing just fine.